Optimization, conditioning and accuracy of radial basis function method for partial differential equations with nonlocal boundary conditions—A case of two-dimensional Poisson equation

Various real-world processes usually can be described by mathematical models consisted of partial differential equations (PDEs) with nonlocal boundary conditions. Therefore, interest in developing computational methods for the solution of such nonclassical differential problems has been growing fast. We use a meshless method based on radial basis functions (RBF) collocation technique for the solution of two-dimensional Poisson equation with nonlocal boundary conditions. The main attention is paid to the influence of nonlocal conditions on the optimal choice of the RBF shape parameters as well as their influence on the conditioning and accuracy of the method. The results of numerical study are presented and discussed.

► The 2D Poisson equation with nonlocal boundary conditions (NBCs) is solved using RBF method. ► Variability of RBF optimal shape parameters respect to parameters of NBCs is investigated. ► The accuracy of results is quite high if optimal or at least near-optimal shape parameters are used. ► The influence of NBCs on the conditioning and accuracy of the method is analyzed. ► The results of numerical study with several test examples are presented and discussed.